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Four essentially different four-square representations of $65$

Example

The integer $65$ has the four-square representations

$65 = 1^{2} + 8^{2} + 0^{2} + 0^{2} = 4^{2} + 7^{2} + 0^{2} + 0^{2} = 2^{2} + 5^{2} + 6^{2} + 0^{2} = 2^{2} + 3^{2} + 4^{2} + 6^{2} .$

Their multisets of absolute values are ${8, 1, 0, 0}$ , ${7, 4, 0, 0}$ , ${6, 5, 2, 0}$ and ${6, 4, 3, 2}$ , which are pairwise distinct, so no two of the four are obtained from one another by permuting coordinates or changing signs: they are essentially different in the sense of Representations as sums of four squares.

Facts & Assumptions

Given: The integer $65$ and the four displayed quadruples.

[F1]

A representation of a nonnegative integer $n$ as a sum of four squares is an ordered quadruple $(a, b, c, d) \in Z^{4}$ with $n = a^{2} + b^{2} + c^{2} + d^{2}$ ; two representations are equivalent up to signs and order exactly when their multisets of absolute values coincide, and essentially different otherwise (Representations as sums of four squares).

[L1]

Every nonnegative integer is a sum of four integer squares (Lagrange's four-square theorem: every nonnegative integer is a sum of four integer squares).

Verification

technique · direct

1.1

Each display is an identity: $1 + 64 + 0 + 0 = 65$ , $16 + 49 + 0 + 0 = 65$ , $4 + 25 + 36 + 0 = 65$ and $4 + 9 + 16 + 36 = 65$ , so all four quadruples are representations of $65$ in the sense of [F1], whose existence [L1] guarantees in advance.

givenF1L1algebra

2.1

The multisets ${8, 1, 0, 0}$ , ${7, 4, 0, 0}$ , ${6, 5, 2, 0}$ and ${6, 4, 3, 2}$ are pairwise distinct, since the first two differ in their largest entry, the third contains $5$ and the others do not, and the fourth is the only one with no entry $0$ ; by the criterion in [F1] the four representations are pairwise essentially different.

step 1.1F1algebra∎

Remarks

Existence and multiplicity are different questions. Lagrange's four-square theorem: every nonnegative integer is a sum of four integer squares asserts that a representation exists. How many essentially different representations a given integer has is not settled by it, and nothing above computes that number for $65$ : the four displayed are exhibited, not claimed to be all.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Keith Conrad, Proofs by Descent, §6, Example 6.1 (standard reference, not scraped)

Four essentially different four-square representations of 6565